3.615 \(\int \frac{1}{x^3 \log ^4(x)} \, dx\)

Optimal. Leaf size=43 \[ -\frac{4}{3} \text{ExpIntegralEi}(-2 \log (x))+\frac{1}{3 x^2 \log ^2(x)}-\frac{1}{3 x^2 \log ^3(x)}-\frac{2}{3 x^2 \log (x)} \]

[Out]

(-4*ExpIntegralEi[-2*Log[x]])/3 - 1/(3*x^2*Log[x]^3) + 1/(3*x^2*Log[x]^2) - 2/(3*x^2*Log[x])

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Rubi [A]  time = 0.0592722, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2306, 2309, 2178} \[ -\frac{4}{3} \text{ExpIntegralEi}(-2 \log (x))+\frac{1}{3 x^2 \log ^2(x)}-\frac{1}{3 x^2 \log ^3(x)}-\frac{2}{3 x^2 \log (x)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Log[x]^4),x]

[Out]

(-4*ExpIntegralEi[-2*Log[x]])/3 - 1/(3*x^2*Log[x]^3) + 1/(3*x^2*Log[x]^2) - 2/(3*x^2*Log[x])

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{1}{x^3 \log ^4(x)} \, dx &=-\frac{1}{3 x^2 \log ^3(x)}-\frac{2}{3} \int \frac{1}{x^3 \log ^3(x)} \, dx\\ &=-\frac{1}{3 x^2 \log ^3(x)}+\frac{1}{3 x^2 \log ^2(x)}+\frac{2}{3} \int \frac{1}{x^3 \log ^2(x)} \, dx\\ &=-\frac{1}{3 x^2 \log ^3(x)}+\frac{1}{3 x^2 \log ^2(x)}-\frac{2}{3 x^2 \log (x)}-\frac{4}{3} \int \frac{1}{x^3 \log (x)} \, dx\\ &=-\frac{1}{3 x^2 \log ^3(x)}+\frac{1}{3 x^2 \log ^2(x)}-\frac{2}{3 x^2 \log (x)}-\frac{4}{3} \operatorname{Subst}\left (\int \frac{e^{-2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac{4}{3} \text{Ei}(-2 \log (x))-\frac{1}{3 x^2 \log ^3(x)}+\frac{1}{3 x^2 \log ^2(x)}-\frac{2}{3 x^2 \log (x)}\\ \end{align*}

Mathematica [A]  time = 0.0163129, size = 43, normalized size = 1. \[ -\frac{4}{3} \text{ExpIntegralEi}(-2 \log (x))+\frac{1}{3 x^2 \log ^2(x)}-\frac{1}{3 x^2 \log ^3(x)}-\frac{2}{3 x^2 \log (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Log[x]^4),x]

[Out]

(-4*ExpIntegralEi[-2*Log[x]])/3 - 1/(3*x^2*Log[x]^3) + 1/(3*x^2*Log[x]^2) - 2/(3*x^2*Log[x])

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Maple [A]  time = 0.005, size = 37, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{x}^{2} \left ( \ln \left ( x \right ) \right ) ^{3}}}+{\frac{1}{3\,{x}^{2} \left ( \ln \left ( x \right ) \right ) ^{2}}}-{\frac{2}{3\,{x}^{2}\ln \left ( x \right ) }}+{\frac{4\,{\it Ei} \left ( 1,2\,\ln \left ( x \right ) \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/ln(x)^4,x)

[Out]

-1/3/x^2/ln(x)^3+1/3/x^2/ln(x)^2-2/3/x^2/ln(x)+4/3*Ei(1,2*ln(x))

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Maxima [A]  time = 1.05405, size = 11, normalized size = 0.26 \begin{align*} -8 \, \Gamma \left (-3, 2 \, \log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/log(x)^4,x, algorithm="maxima")

[Out]

-8*gamma(-3, 2*log(x))

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Fricas [A]  time = 2.04585, size = 115, normalized size = 2.67 \begin{align*} -\frac{4 \, x^{2} \log \left (x\right )^{3} \logintegral \left (\frac{1}{x^{2}}\right ) + 2 \, \log \left (x\right )^{2} - \log \left (x\right ) + 1}{3 \, x^{2} \log \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/log(x)^4,x, algorithm="fricas")

[Out]

-1/3*(4*x^2*log(x)^3*log_integral(x^(-2)) + 2*log(x)^2 - log(x) + 1)/(x^2*log(x)^3)

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Sympy [A]  time = 0.675919, size = 32, normalized size = 0.74 \begin{align*} - \frac{4 \operatorname{Ei}{\left (- 2 \log{\left (x \right )} \right )}}{3} + \frac{- 2 \log{\left (x \right )}^{2} + \log{\left (x \right )} - 1}{3 x^{2} \log{\left (x \right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/ln(x)**4,x)

[Out]

-4*Ei(-2*log(x))/3 + (-2*log(x)**2 + log(x) - 1)/(3*x**2*log(x)**3)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \log \left (x\right )^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/log(x)^4,x, algorithm="giac")

[Out]

integrate(1/(x^3*log(x)^4), x)